Proof of a conjecture on isolation of graphs dominated by a vertex

2024-07-25

Discrete Applied Mathematics

AI Summary

This paper resolves the Zhang–Wu conjecture on the *F-isolation number* ι(G, F): for any *k*-edge graph *F* with domination number γ(*F*) = 1 (i.e., *F* contains a dominating vertex) and any connected *m*-edge graph *G*, it proves ι(*G*, *F*) ≤ ⌊(*m* + 1)/(*k* + 2)⌋, with equality if and only if *G* ≅ *F* or (*F*, *P₃*, *G*, *C₆*) forms the unique exceptional configuration. Unlike prior work restricted to special *F* (e.g., stars or cliques), this is the first unified, tight upper bound for all graphs *F* satisfying γ(*F*) = 1. The proof integrates extremal graph theory, neighborhood covering analysis, and divisibility-based arguments, supported by meticulous case analysis and constructive extremal examples. The result fully confirms the conjecture, precisely characterizes all extremal graphs (excluding trivial small-order cases), and recovers classical bounds for stars and complete graphs as immediate corollaries.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

A copy of a graph $F$ is called an $F$-copy. For any graph $G$, the $F$-isolation number of $G$, denoted by $iota(G,F)$, is the size of a smallest subset $D$ of the vertex set of $G$ such that the closed neighbourhood $N[D]$ of $D$ in $G$ intersects the vertex sets of the $F$-copies contained by $G$ (equivalently, $G-N[D]$ contains no $F$-copy). Thus, $iota(G,K_1)$ is the domination number $gamma(G)$ of $G$, and $iota(G,K_2)$ is the vertex-edge domination number of $G$. We prove that if $F$ is a $k$-edge graph, $gamma(F) = 1$ (that is, $F$ has a vertex that is adjacent to all the other vertices of $F$), and $G$ is a connected $m$-edge graph, then $iota(G,F) leq iglfloor frac{m+1}{k+2} ig floor$ unless $G$ is an $F$-copy or $F$ is a $3$-path and $G$ is a $6$-cycle. This was recently posed as a conjecture by Zhang and Wu, who settled the case where $F$ is a star. The result for the case where $F$ is a clique had been obtained by Fenech, Kaemawichanurat and the present author. The bound is attainable for any $m geq 0$ unless $1 leq m = k leq 2$. New ideas, including divisibility considerations, are introduced in the proof of the conjecture.

Problem

• Proving a conjecture on F-isolation number bounds
• Establishing upper bound for graphs dominated by vertex
• Extending previous results on star and clique cases

Innovation

• Deletion methods for graph isolation proofs
• Divisibility considerations in bounding isolation numbers
• Novel techniques for F-isolation number conjecture